Hidden Markov Models

An elementary first-order Hidden Markov model is a probabilistic model over $N$ observations, $y_n$, and $N$ hidden states, $x_n$, which can be fully defined by the conditional distributions $p(y_n\mid x_n,\varphi )$ and $p(x_n\mid x_{n-1},\varphi )$. Here we make the dependency on additional model parameters, $\varphi$, explicit. When $x$ is continuous, the user can explicitly encode these distributions in Stan and use Markov chain Monte Carlo to integrate $x$ out.

When each state $x$ takes a value over a discrete and finite set, say $\{1,2,...,K\}$, we can take advantage of the dependency structure to marginalize $x$ and compute $p(y\mid \varphi )$. We start by defining the conditional observational distribution, stored in a $K\times N$ matrix $\omega$ with $${\omega }_{kn}=p(y_n\mid x_n=k,\varphi ).$$ Next, we introduce the $K\times K$ transition matrix, $\mathrm{\Gamma }$, with $${\mathrm{\Gamma }}_{ij}=p(x_n=j\mid x_{n-1}=i,\varphi ).$$ Each row defines a probability distribution and must therefore be a simplex (i.e. its components must add to 1). Currently, Stan only supports stationary transitions where a single transition matrix is used for all transitions. Finally we define the initial state $K$-vector $\rho$, with $${\rho }_k=p(x_0=k\mid \varphi ).$$

The Stan functions that support this type of model are special in that the user does not explicitly pass $y$ and $\varphi$ as arguments. Instead, the user passes $\log \omega$, $\mathrm{\Gamma }$, and $\rho$, which in turn depend on $y$ and $\varphi$.

Stan functions

real hmm_marginal(matrix log_omega, matrix Gamma, vector rho)
Returns the log probability density of $y$, with $x_n$ integrated out at each iteration.

Available since 2.24

The arguments represent (1) the log density of each output, (2) the transition matrix, and (3) the initial state vector.

• log_omega: $\log {\omega }_{kn}=\log p(y_n\mid x_n=k,\varphi )$, log density of each output,

• Gamma: ${\mathrm{\Gamma }}_{ij}=p(x_n=j|x_{n-1}=i,\varphi )$, the transition matrix,

• rho: ${\rho }_k=p(x_0=k\mid \varphi )$, the initial state probability.

array[] int hmm_latent_rng(matrix log_omega, matrix Gamma, vector rho)
Returns a length $N$ array of integers over $\{1,...,K\}$, sampled from the joint posterior distribution of the hidden states, $p(x\mid \varphi ,y)$. May be only used in transformed data and generated quantities.

Available since 2.24

matrix hmm_hidden_state_prob(matrix log_omega, matrix Gamma, vector rho)
Returns the matrix of marginal posterior probabilities of each hidden state value. This will be a $K\times N$ matrix. The $n^{\mathrm{th}}$ column is a simplex of probabilities for the $n^{\mathrm{th}}$ variable. Moreover, let $A$ be the output. Then $A_{ij}=p(x_j=i\mid \varphi ,y)$. This function may only be used in transformed data and generated quantities.

Available since 2.24